Why are there 2 pi radians in a circle?

1 Answer
Jim H
Aug 30, 2015

It is not arbitrary, but it is because of the definition of radian measure.

Explanation:

One definition says the radian measure of angle #theta# is the ratio of arc length #s# to radius #r# when #theta# is made central in a circle of radius #r#.

The arc length of the whole circle is its circumference #2pir#, so the angle that goes once around the circle has radian measure #C/r = (2pir)/r = 2pi#

This definition is the reason that we have the formula for arc length #s = r theta# as long as we measure #theta# in radians

Note that whatever unit are used to measure #r# and #s#, in the definition of radian measure they cancel. We say that radian measure is dimesionless.

Also note that some teachers introduce radian measure without discussing the 'official' definition as a ratio of lengths. They simply announce that once around the circle is #2pi# radians. (Some say #360^@# is the same as #2pi# radians.)
This makes it look arbitrary when it is not arbitrary at all.

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